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# A topological splitting theorem for Poincar\'e duality groups and high dimensional manifolds

Record type: Article

Waldhausen's celebrated torus theorem plays a central role in the classification of topological 3-manifolds. It also led to a number of algebraic splitting theorems for discrete groups including Kropholler's algebraic torus theorem for Poincar\'e duality groups and to the algebraic annulus theorems of Dunwoody/Sageev and Scott/Swarup. Here, in the same spirit, we offer topological and algebraic decomposition theorems in the context of high dimensional aspherical manifolds, providing an algebraic splitting theorem for Poincar\'e duality groups and exploiting Cappell's splitting theory to extract the required topological splittings. As a result we show that for a wide class of manifold pairs $N,M$ with $\dim(M)=\dim(N)+1$, every, $\pi_1$-injective map f$N\rightarrow M$ factorises up to homotopy as a finite cover of an embedding. As an application of this we show that under certain circumstances the vanishing of the first Betti number for $M$ is an obstruction to the existence of such maps.

PDF SuperRigidity11102011(ArXiv_updated_version).pdf - Other
PDF 1110.2041v3.pdf - Other
PDF 1110.2041v3.pdf - Accepted Manuscript

## Citation

Niblo, Graham and Kar, Aditi (2013) A topological splitting theorem for Poincar\'e duality groups and high dimensional manifolds Geometry & Topology, 17, pp. 2203-2221.

Submitted date: 28 July 2010
Accepted/In Press date: 26 April 2013
Published date: 26 July 2013
Keywords: Torus theorem, Poincaré duality group, Bass-Serre theory, Kazhdan's property (T), Borel conjecture, surgery, Cappell's splitting theorem, embeddings, rigidity, geometric group theory, quaternionic hyperbolic manifolds
Organisations: Pure Mathematics

## Identifiers

Local EPrints ID: 161381
URI: http://eprints.soton.ac.uk/id/eprint/161381
ISSN: 1465-3060
PURE UUID: a971c878-b841-4f85-9d94-624680eaef9a

## Catalogue record

Date deposited: 28 Jul 2010 18:59

## Contributors

Author: Graham Niblo